Have you tried chiropractic?
I have your same symptoms at my left pelvis, it does feel like something is being pulled down by gravity, it's like having an extra weight on my left side.
I have been treated by a chiropractor twice so far in the last 2 weeks and the pain has decreased dramatically. I got explained and showed how my left pelvis was shifted upwards effecting tissues and muscles around it.
My chiropractor managed to shift my pelvis back in position, however every 1 or 2 weeks I have to go back to make sure it is still in place as between sessions the pelvis tends to move back to the incorrect position.
The day after the treatments some pain is getting out due to the movement of the bones, the second day you will feel the improvements.
I am combining chiropractic with physiotherapy to remove the pain to the tissues and muscles around the pelvis removing the pain in the tissues in my butt chins.
physiotherapy is an excellent pain relief but you will keep getting pains if you don't fix the pelvis as the pelvis in the wrong position causes tissues and muscles to stretch.
Personally at first I didn't know what my pain was cause from so I started with physiotherapy, 1 session every week for 3 weeks. After that I was in heaven however my physiotherapist suggested me to visit a chiropractic as she thought my pelvis was causing the issue (spot on!). I have now visited my chiropractor twice in the last 2 weeks and now asked to wait a bit longer between sessions as the pelvis will get better and better, but it takes time.